Uniformity Transition for Ray Intensities in Random Media

This paper analyses a model for the intensity of distribution for rays propagating without absorption in a random medium. The random medium is modelled as a dynamical map. After $N$ iterations, the intensity is modelled as a sum $S$ of $\mathcal{N}$ contributions from different trajectories, each of which is a product of $N$ independent identically distributed random variables $x_k$, representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: $\mathcal{N}={\mathrm{\Lambda }}^N$, for some $\mathrm{\Lambda }>1$. We investigate the probability distribution of $S$. We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of $S$ are suppressed as $N\to \mathrm{\infty }$, and a phase where the $S$ has large fluctuations, for which we provide a large deviation analysis.